# Smooth morphism

In algebraic geometry, a morphism $f:X\to S$ between schemes is said to be smooth if

(iii) means that for any $s\in S$ the fiber $f^{-1}(s)$ is a nonsingular variety. Thus, intuitively speaking, a smooth morphism gives a flat family of nonsingular varieties.

If S is the spectrum of a field and f is of finite type, then one recovers the definition of a nonsingular variety.

There are many equivalent definitions of a smooth morphism. Let $f:X\to S$ be locally of finite presentation. Then the following are equivalent.

A morphism of finite type is étale if and only if it is smooth and quasi-finite.

A smooth morphism is stable under base change and composition. A smooth morphism is locally of finite presentation.

A smooth morphism is universally locally acyclic.

## Formally smooth morphism

{{#invoke:see also|seealso}} One can define smoothness without reference to geometry. We say that a S-scheme X is formally smooth if for any affine S-scheme T and a subscheme $T_{0}$ of T given by a nilpotent ideal, $X(T)\to X(T_{0})$ is surjective where we wrote $X(T)=\operatorname {Hom} _{S}(T,X)$ . Then a morphism locally of finite type is smooth if and only if it is formally smooth.

In the definition of "formally smooth", if we replace surjective by "bijective" (resp. "injective"), then we get the definition of formally étale (resp. formally unramified).